Proving that the following is an exact sequence

Question

If $L$ and $N$ are submodules of $M$ show that the following is an exact sequence:

$$ 0\to M/(L\cap N) \to (M/L)\oplus (M/N) \to M/(L+N )\to 0 $$

for the last morphism $g(x +L , y+N) = (x-y) + (L+N)$, what about the first? And should I use that $$ 0 \to I\cap J\to I\oplus J\to I+J\to 0 $$ is already a short exact sequence?? Thanks :)

Answer 1

The first homomorphism is natural: $(x +L\cap N)\mapsto (x + L, x+ N)$